20 Activity 2

The purpose of this activity is to explore and reinforce concepts in Chapters 1-3 of Bringing Bayesian Models to Life.

You are encouraged to work with others students in the class to complete this activity.

For this activity let $[a]$ be a beta distribution with $\alpha=2$ and $\beta=1$ (i.e., $a\sim\text{beta}(\alpha=2,\beta=1)$).

1. Solve the integral $\int_{0}^1 [a]da$ analytically (i.e., using pencil and paper).

2. Approximate the integral $\int_{0}^1 [a]da$ numerically using the quadrature method with $m=40$ equally spaced support points (see pg. 12 in BBM2L).

3. Approximate the integral $\int_{0}^1 [a]da$ using Monte Carlo integration with $K=100$ draws (see pg. 18 in BBM2L).

4. Write 3-5 sentences comparing the results you obtained in questions 1-3 and explain why the analytical and numerical results differ.

5. Solve the integral $\int_{0}^1 a[a]da$ analytically. Verify that your answer matches that of the expected value of a beta distribution with $\alpha=2$ and $\beta=1$.

6. Approximate the integral $\int_{0}^1 a[a]da$ numerically using the quadrature method with $m=40$ equally spaced support points.

7. Approximate the integral $\int_{0}^1 a[a]da$ using Monte Carlo integration with $K=100$ draws.

8. This is probably a trick question. Solve the integral $\int_{0}^1 \text{log}\left(\frac{1}{\mathrm{sin}(a)}\right)[a]da$ analytically.

9. This is not a trick question. Approximate the integral $\int_{0}^1 \text{log}\left(\frac{1}{\mathrm{sin}(a)}\right)[a]da$ using Monte Carlo integration with $K=100$ draws.